Properties

Label 2-3450-1.1-c1-0-42
Degree $2$
Conductor $3450$
Sign $1$
Analytic cond. $27.5483$
Root an. cond. $5.24865$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 2·11-s + 12-s + 2·13-s + 2·14-s + 16-s + 18-s + 8·19-s + 2·21-s + 2·22-s + 23-s + 24-s + 2·26-s + 27-s + 2·28-s − 10·29-s + 8·31-s + 32-s + 2·33-s + 36-s − 8·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.235·18-s + 1.83·19-s + 0.436·21-s + 0.426·22-s + 0.208·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.377·28-s − 1.85·29-s + 1.43·31-s + 0.176·32-s + 0.348·33-s + 1/6·36-s − 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(27.5483\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.557226263\)
\(L(\frac12)\) \(\approx\) \(4.557226263\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 \)
23 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.330679512375419974632141222812, −7.972657229440826456735322991843, −6.99676669156697235971803371524, −6.45378015064161628438021729783, −5.22663091374375628833076327280, −4.96062417466256580093711065106, −3.62036349522950108759759526013, −3.40767042282619928027437921673, −2.04320074130275352163442800889, −1.26444559536152139973964905961, 1.26444559536152139973964905961, 2.04320074130275352163442800889, 3.40767042282619928027437921673, 3.62036349522950108759759526013, 4.96062417466256580093711065106, 5.22663091374375628833076327280, 6.45378015064161628438021729783, 6.99676669156697235971803371524, 7.972657229440826456735322991843, 8.330679512375419974632141222812

Graph of the $Z$-function along the critical line